Learned barrier certificates provide a principled mechanism for establishing safety of stochastic systems, but their validity may deteriorate during deployment when the underlying dynamics differ from those used for training. We propose an anytime-valid runtime monitor that audits a learned barrier certificate using observed closed-loop trajectories. The monitor is intended to detect loss of statistical support for the deployed barrier condition; it does not by itself guarantee detection before unsafe entry or safety of a subsequent intervention. Specifically, the conditional barrier-drift condition is formulated as an episode-wise sequential hypothesis test, and observed barrier increments are transformed into a nonnegative test supermartingale, yielding finite-sample false-alarm control under arbitrary stopping times. Inspired by finite-horizon safety verification, we further introduce a deterministic temporal shaping of the monitoring boundary, making the monitor more conservative early in a prescribed deployment horizon while preserving anytime-validity. The monitor is designed to detect statistical evidence that the safety certificate is no longer consistent with the observed dynamics, rather than to predict unsafe entry. To assess the monitor empirically, we evaluate empirical alarm behavior under valid and invalid certificates. Code and data are available at https://github.com/FormalCert-AI/runtime-barrier-monitor.
Steinke and Zakynthinou(2020) introduces the Conditional Mutual Information (CMI) framework of analyzing the information complexity of learning algorithms based on algorithm-dependent information-theoretic quantities. We study one of these quantities, the evaluated Conditional Mutual Information (eCMI). It has been an interesting question whether the optimal PAC guarantee for VC classes can be recovered from the algorithm-dependent analyses via CMI. And we show that it is possible to recover this guarantee by constructing a learning algorithm whose eCMI is of order O(d) in the realizable case, where d is the VC-dimension of the concept class. Specially, our algorithm is a randomized Majority-of-5 base learners with optimal in-expectation generalization guarantee.
Knowledge graph learning offers a powerful framework for representing and inferring structured knowledge, with broad practical applications. Two obstacles limit current methods: labeled triples are scarce for each entity and relation, which makes expressive models hard to train, and scoring functions are designed ad hoc, which limits their generalizability and leaves them without theoretical grounding. We address both obstacles with a theoretically grounded framework, with end-to-end guarantees, that extends and subsumes existing methods. The framework proceeds in two stages: multi-view unsupervised pretraining that integrates heterogeneous sources of entity side information into a shared representation, followed by supervised learning across multiple relation types. We establish a unified nonasymptotic risk bound that combines the subspace recovery error of pretraining and the sample complexity of the labeled data in a single guarantee, quantifying formally how large-scale unlabeled data from multiple views benefits downstream knowledge prediction. Synthetic experiments validate each component of the theory, and experiments on large-scale real-world knowledge graph benchmarks confirm the effectiveness of our approach.
We prove the population-level identifiability of deep generative models (DGMs) with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors, in a purely unsupervised setting. We introduce three algebraic contrast principles for symmetry breaking: *domain contrast*, which trivializes the mixture symmetry group, a label-free counterpart of the sufficient-variability assumption of nonlinear ICA; *mechanism contrast*, which ensures every decoder branch gets an observational boundary across which it toggles alone; and *interaction contrast*, which forbids parameter conspiracies between latent components and decoder branches. Together they exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetry structure of the latent GMM. Continuity is replaced by algebraic symmetry conditions; injectivity is decoupled from structural identification and required only for pointwise inversion. Our results form a hierarchy: from *law identifiability* (LID; latent distribution up to a global affine map) through *map identifiability* (MID; decoder up to the same map) to *posterior* and *pointwise identifiability*. The ICA-form ambiguity emerges under conditions on diagonal component covariances. The interaction contrast is the only one of our contrasts required on the estimator side. To our knowledge this is the first to develop algebraic symmetry-breaking into an identifiability engine for a concrete nonlinear model class, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.
Digital Twins support real-time decision-making through predictive models of physical systems, yet their reliability can deteriorate as operating conditions evolve. Maintaining predictive fidelity under such changes is challenging, particularly when model adaptation relies on limited streaming data, and updated models must be evaluated before deployment. Here, we present an online adaptation framework that integrates gradient-based drift detection, parameter-efficient fine-tuning, and statistical model validation within model predictive control. Changes in prediction-loss gradients trigger batch-wise model updates using Low-Rank Adaptation, while a paired moving-block bootstrap test evaluates predictive improvement before deploying the updated model. The framework is evaluated using a stochastic dynamical system with abrupt changes and a simulated directed energy deposition process with gradual changes in material properties. Results show improvements in predictive accuracy and uncertainty estimation during the evaluated post-adaptation intervals, alongside improved reference tracking and reduced constraint violations in the manufacturing case study compared with the investigated baselines. These findings demonstrate the potential of integrating continual validation with model adaptation while acknowledging the trade-off between responsiveness and predictive reliability in evolving Digital Twins.
Many real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG). In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depths at which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by \citep{dunlop2018} on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theory and methodology at scale and under DAG misspecification. We then model a latent-collider DAG, a protein signalling network, and a multi-fidelity \mbox{heavy-ion} collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability over the simulator hierarchy.
We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.
Clustering is a central tool for discovering latent structure in unlabeled data; yet modern clustering pipelines often end with a hard assignment of each observation to a cluster without rigorous measures of assignment uncertainty. We propose a novel weighted conformal approach for constructing valid confidence sets for cluster labels. The key difficulty is that the labels available for calibration are not observed ground-truth labels, but synthetic labels produced by a data-dependent clustering algorithm. Our method develops a conformal inference algorithm that corrects the resulting mismatch with the latent target labels through weights by formulating conformal clustering as a conditional label-distribution shift problem. We first derive an oracle procedure that attains finite-sample marginal coverage and then develop a computationally tractable and implementable version using estimated conditional label probabilities and novel augmented calibration. We show that the coverage of the estimated-weight procedure depends on the estimator, giving an explicit bound on the loss relative to the nominal level. Empirical studies demonstrate that the proposed weighted approach offers improvements over the recently proposed split conformal clustering procedure in terms of informative confidence set size, especially in nonlinear and high-dimensional clustering applications.
Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent sampling. We consider strong mixing dependent observations and ask whether dependence changes the scale of nearest-neighbor neighborhoods. We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension, making the results applicable to high-dimensional data supported on lower-dimensional manifolds. Synthetic experiments and real-world time-series benchmarks illustrate the theory, showing that nearest-neighbor geometry remains informative under dependent sampling.
Distributional causal inference requires estimating not only average treatment effects but also interventional outcome distributions, including quantiles, tail risks, and policy-dependent uncertainty. As a method for distributional causal inference, generative adversarial network (GAN)-based counterfactual methods are flexible tools for this task. However, these methods have several limitations. First, the objectives of certain techniques do not coincide with the statistical risk of the identifiable causal target, and therefore provide limited theoretical guarantees regarding estimable counterfactual distributions or optimality. Second, implementing target-design risks under observational sampling can require density-ratio estimation. In this paper, we propose GANICE (GAN for Interventional Conditional Estimation) with several advantages: it (i) specifies the conditional interventional distribution for each treatment--covariate state as the causal estimation target; (ii) estimates the conditional distribution such that its averaged Wasserstein risk is minimized; (iii) attains minimax optimality. GANICE achieves these advantages through the extended Wasserstein distance and cell-normalized estimation, with minimax bounds based on Wasserstein GAN oracle inequalities and an anisotropic Assouad construction; Besov theory characterizes the cellwise critics. Our experiments show that GANICE outperforms existing methods on most benchmarks.
\emph{Kullback-Leibler} (KL) regularization is ubiquitous in reinforcement learning algorithms in the form of \emph{reverse} or \emph{forward} KL. Recent studies have demonstrated $\epsilon^{-1}$-type fast rates for decision making under reverse KL regularization, in contrast to the standard $\epsilon^{-2}$-type sample complexity. However, for forward-KL-regularized objectives, existing statistical analyses are either not applicable or result in $\tilde{O}(\epsilon^{-2})$ slow rates. We take the first step towards addressing this problem via a streamlined analysis of forward-KL-regularized offline CBs. We give the first $\tilde{O}(\epsilon^{-1})$ upper bounds in tabular and general function approximation settings, both under notions of \emph{single-policy concentrability}. In particular, our convex-analytical pipeline unifies these settings by exploiting the pessimism principle in a novel way and completely bypasses the proof routines in previous works based on the mean value theorem, which might be of independent interest. Moreover, we provide rate-optimal lower bounds, manifesting the tightness of our upper bounds in terms of statistical rates. Our lower bounds also demonstrate that the forward-KL-regularized sample complexity recovers the unregularized slow rate in the low-regularization regime, similarly to the reverse-KL regularization.
We address the brittleness of Bayesian experimental design under model misspecification by formulating the problem as a max--min game between the experimenter and an adversarial nature subject to information-theoretic constraints. We demonstrate that this approach yields a robust objective governed by Sibson's $\alpha$-mutual information~(MI), which identifies the $\alpha$-tilted posterior as the robust belief update and establishes the R\'enyi divergence as the appropriate measure of conditional information gain. To mitigate the bias and variance of nested Monte Carlo estimators needed to estimate Sibson's $\alpha$-MI, we adopt a PAC-Bayes framework to search over stochastic design policies, yielding rigorous high-probability lower bounds on the robust expected information gain that explicitly control finite-sample error.
We study conditional average treatment effect (CATE) estimation from observational data in regimes with potentially weak overlap, when the nuisance response functions lie in a reproducing kernel Hilbert space (RKHS), but the contrast function is structurally simpler. We consider three forms of such simplicity: (i) membership in a lower-complexity RKHS with faster eigenvalue decay, (ii) a source condition relative to the nuisance kernel, or (iii) dependence on a known low-dimensional covariate representation. We develop a unified two-stage kernel ridge regression procedure that remains valid under weak overlap and attains minimax rates determined by the complexity of the contrast function. The procedure avoids propensity-score estimation and separate nuisance convergence-rate assumptions. We also show that a simple model-selection step over candidate contrast spaces and regularization levels satisfies an oracle inequality, enabling adaptation to unknown CATE regularity.
Recovering a low-CP-rank tensor from noisy linear measurements is a central challenge in high-dimensional data analysis, with applications spanning tensor PCA, scalar-on-tensor regression, tensor-on-tensor regression, and beyond. We exploit the intrinsic geometry of rank-one tensors by casting the recovery task as an optimization problem over the Segre manifold, the smooth Riemannian manifold of rank-one tensors. This geometric viewpoint yields two powerful algorithms: Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN), each of which preserves feasibility at every iteration. Under mild noise assumptions, we prove that RGD converges at a local linear rate, while RGN exhibits an initial local quadratic phase followed by linear refinement to the statistical error. Extensive synthetic experiments validate these convergence guarantees and demonstrate the practical effectiveness of our methods.
We study kernel-based learning of analytic operators from finite samples, with a focus on how the regularity of the target should determine the choice of the kernel. We establish embeddings of analytic operators into vector-valued reproducing kernel Hilbert spaces and derive finite-sample, high-probability guarantees that quantify the uncertainty due to random sampling: in the noiseless setting, suitable analytic kernels yield rates exponential in a fractional power of the sample size, while Sobolev-type kernels yield polynomial rates. We apply the theory to the solution operator of a parametric elliptic PDE, where the training data are themselves finite-element approximations, and obtain bounds that separate the sampling and discretization contributions to the error and prescribe how the mesh size should be coupled to the sample size. The results are illustrated numerically.
Reward models used for off-policy evaluation (OPE) are commonly selected using held-out data. When sample size and context dimension grow proportionally, the difference in test risk between models fitted on the full sample and on a fixed fraction of it need not vanish. Held-out evaluation can therefore consistently estimate the test risk of the model trained on the smaller sample, but because the relative ranking of candidate reward models can change after full-data refitting, held-out selection can be suboptimal. We develop an Onsager-corrected, target-policy-weighted criterion that estimates the full-data reward model's out-of-sample risk under the target policy using only its training observations. This enables early stopping along the full-data gradient-descent trajectory without sample splitting. For neural networks with fixed depth and hidden width under Gaussian contexts, single-index rewards, context-dependent multi-index target policies, and known context-independent logging, we prove uniform consistency of our criterion over a fixed finite set of gradient-descent iterates and a near-oracle guarantee for the selected reward model. The same correction yields a doubly robust estimator that consistently estimates the true policy value without sample splitting, even when reward-model prediction error does not vanish. Experiments show decreasing risk-estimation error as dimension grows at a fixed sample-to-dimension ratio, and improved reward-model selection relative to held-out baselines. Empirically, our correction also reduces the root mean squared error of doubly robust value estimates computed from the same data used to fit and select the reward models.
Low-rank approximation is a basic tool for inference from large and noisy data matrices. However, when noise variances differ across entries, a setting known as heteroskedasticity, the noise can distort the observed spectrum, producing spurious components and obscuring weak signals. To address these challenges, we develop BRINE (Blind Resolvent-based Iterative Noise Equalizer), a data-driven procedure that iteratively rescales rows and columns to equalize the average noise variance across both dimensions. Under suitable conditions, this procedure restores the Marchenko-Pastur law for the noise spectrum and can amplify signal components without prior knowledge of the signal or noise structure. A key idea underlying our approach is to equalize the diagonal entries of the data resolvent as a proxy for equalizing the unknown noise variances. At the population level, we identify a potential function that characterizes this equalization and motivates an iterative minimization procedure. We prove convergence of the iterations to the true equalizer for arbitrary positive variance profiles, with a geometric rate under mild conditions. The updates can be estimated directly from noisy data, and we establish probabilistic error bounds for the scaling factors. Simulations and experiments across diverse biological data modalities and preprocessing pipelines illustrate the effectiveness of our approach.
We study binary classification when the two classes are elliptical with a shared radial generator $g$. The log-likelihood ratio is then an additive function of the two squared Mahalanobis radii, with radial link $\varphi=\log g$. For Gaussian classes $\varphi$ is affine and the rule reduces to quadratic discriminant analysis (QDA); otherwise the Bayes link is not the identity. Instead of fitting this link with a spline smoother, we derive its family from the within-class law of the radius and estimate a finite fractional-power projection. We prove that the link is identifiable from the radius law, that the plug-in estimator is $\sqrt{n}$-consistent and asymptotically normal under finite-moment conditions, and that the induced classifier is asymptotically Bayes-optimal in an iterated sieve limit. For $d\ge2$ and non-nested class parameters, the identity link suffices only for affine (Gaussian) generators. The bridge, GAM membership and the affine direction of this dichotomy are verified in Lean 4. We compare against the global Mahalanobis-GAM of Ghosh et al. (2025), re-implemented with mgcv REML splines at equal input budget. On two UCI benchmarks the derived link ties with the fitted GAM. On six financial series from three asset classes, with validation that respects temporal dependence, it is never significantly worse than the fitted GAM: better on CAD/USD and tied on the other five. It beats QDA on the three heaviest-tailed series (oil $[+0.023,+0.069]$, S&P 500 $[+0.039,+0.125]$, JPY/USD $[+0.009,+0.047]$) and ties elsewhere. A rate simulation with a closed-form ground truth reproduces the $\sqrt{n}$ rate and, under covariance heterogeneity, QDA's risk floor against the derived link's vanishing excess risk. The derived link thus matches a tuned GAM without smoothing-parameter selection and improves on QDA where the generator departs from the Gaussian.
Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as budgeted heteroskedastic multi-judge estimation under independent, unbiased score observations. Given $K$ prompt-response pairs and $J$ judges with known costs but unknown query-judge variances, our goal is to estimate the bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its leading error bound. Since this allocation depends on the unknown variances, we then address the unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and uses optimistic variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a local minimax lower bound matching the leading allocation-dependent rate, establishing asymptotic instance-optimality up to logarithmic factors. An Assouad-type construction based on local, variance-preserving perturbations yields this bound both in expectation and with constant probability. A complementary Fano-type change-of-measure argument captures the logarithmic dependence on the confidence level with a different (suboptimal) allocation geometry. Finally, we numerically validate our approach on synthetic data and resampled HelpSteer2 evaluations, demonstrating improvements over na\"ive uniform allocation.
Self-distillation has emerged as a promising technique for improving model performance in modern machine learning systems. We develop the statistical foundations of self-distillation in high-dimensional linear regression by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We consider both spiked and general covariance matrices. For the former, we establish that when the covariance has $s$ distinct spikes, $s$-step self-distillation achieves optimal prediction among spectral shrinkage estimators, outperforming well-known estimators in statistics and machine learning. Moreover, we show that $s$ steps are necessary for optimality: any $(s-k)$-step distilled estimator is strictly suboptimal for $1 \leq k \leq s$. For general covariance matrices, we show that self-distillation approaches the optimal limiting prediction risk among continuous spectral shrinkage estimators as the number of distillation steps increases. We further develop a generalized cross-validation procedure for selecting the distillation parameters from data and establish an asymptotic oracle guarantee for the resulting estimator. We also study a federated approach where multiple data centers share spectral shrinkage estimators and a common server seeks to aggregate them to achieve optimal performance. In this case, we find that the best local rule again takes the form of self-distillation, though it differs from the optimal rule when data are hosted centrally on a single server. Together, our results explain why self-distillation improves predictive performance and provide a broader statistical framework that connects it to classical shrinkage-based methods.